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arXiv:1707.02904 · Physical Review A

Impurity states in the one-dimensional Bose gas

V. Pastukhov

Published Oct 25, 2017• 27 citations
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Abstract

The detailed study of the low-energy spectrum for a mobile impurity in the one-dimensional bosonic environment is performed. Particularly we have considered only two analytically accessible limits, namely, the case of an impurity immersed in a dilute Bose gas where one can use many-body perturbative techniques for low-dimensional bosonic systems and the case of the Tonks-Girardeau gas for which the usual fermionic diagrammatic expansion up to the second order is applied.

DOI
10.1103/physreva.96.043625

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The detailed study of the low-energy spectrum for a mobile impurity in the one-dimensional bosonic environment is performed. Particularly we have considered only two analytically accessible limits, namely, the case of an impurity immersed in a dilute Bose gas where one can use many-body perturbative techniques for low-dimensional bosonic systems and the case of the Tonks-Girardeau gas for which the usual fermionic diagrammatic expansion up to the second order is applied.
A few years ago, flow equations were introduced as a technique for calculating the ground-state energies of cold Bose gases with and without impurities[1,2]. In this paper, we extend this approach to compute observables other than the energy. As an example, we calculate the densities, and phase fluctuations of one-dimensional Bose gases with one and two impurities. For a single mobile impurity, we use flow equations to validate the mean-field results obtained upon the Lee-Low-Pines transformation. We show that the mean-field approximation is accurate for all values of the boson-impurity interaction strength as long as the phase coherence length is much larger than the healing length of the condensate. For two static impurities, we calculate impurity-impurity interactions induced by the Bose gas. We find that leading order perturbation theory fails when boson-impurity interactions are stronger than boson-boson interactions. The mean-field approximation reproduces the flow equation results for all values of the boson-impurity interaction strength as long as boson-boson interactions are weak.
We discuss impurities in a one-dimensional Bose gas with arbitrary boson-boson and boson-impurity interactions. To fully account for quantum effects, we employ numerical simulations based on the density-matrix renormalization group (DMRG) and—in the regime of strong boson-boson interactions—the mapping to weakly interacting fermions. A mean-field description of the Bose polaron based on coupled Gross–Pitaevskii Schrödinger equations predicts the existence of a self-localized polaron. We here show that such a solution does not exist and is an artifact of the underlying decoupling approximation. To this end we consider a mobile impurity in a box potential. Our work demonstrates that correlations between the impurity position and the bosons are important even in the limit where mean-field approaches are expected to work well. Furthermore, we derive analytical approximations for the energy of a single polaron formed by a heavy impurity for arbitrary interaction strengths and large but finite boson-boson couplings which accurately reproduce DMRG results. This demonstrates that the polaron problem of a heavy impurity in a one-dimensional Bose gas can be accurately approximated by a proper mean-field description plus a linearized treatment of quantum fluctuations for arbitrary boson-boson and impurity-boson couplings. Finally, we determine the polaron-polaron interaction potential V ( r ) in the Born-Oppenheimer approximation for small and intermediate distances r , which in the Tonks gas limit is oscillatory due to Friedel oscillations in the Bose gas.
The weakly interacting Bose gas in two dimensions is considered in the dilute limit n1/2{n}^{1/2}n1/2a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is Tc{T}_{c}Tc​\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/na2{\mathrm{na}}^{2}na2)], and the condition of validity of the dilute limit is ln ln(1/na2{\mathrm{na}}^{2}na2)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.
We study two mobile bosonic impurities immersed in a one-dimensional optical lattice and interacting with a bosonic bath. We employ the exact diagonalization method for small periodic lattices to study stationary properties and dynamics. We consider the branch of repulsive interactions that induce the formation of bound impurities, akin to the bipolaron problem. A comprehensive study of ground-state and low-energy properties is presented, including an examination of the interaction strengths which induce the formation of a bound dimer of impurities. We also study the dynamics induced after an interaction quench to examine the stability of the bound dimers. We reveal that after large interaction quenches from strong to weak interactions the system can show large oscillations over time with revivals of the dimer states. We find that the oscillations are driven by selected eigenstates with phase-separated configurations.
Abstract The properties of an impurity immersed in a dilute D -dimensional Bose gas at temperatures close to its second-order phase transition point are considered. Particularly by means of the 1/ N -expansion, we calculate the leading-order polaron energy and the damping rate in the limit of vanishing boson–boson interaction. It is shown that the perturbative effective mass and the quasiparticle residue diverge logarithmically in the long-length limit, signalling the non-analytic behavior of the impurity spectrum and pole-free structure of the polaron Green’s function in the infrared region, respectively.